30June2000 16:17

A HYBRID FINITE VOLUME/PDF MONTE CARLO METHOD TO CAPTURE SHARPGRADIENTS IN UNSTRUCTURED GRIDS

Genong Li and Michael F. ModestMechanical Engineering DepartmentThe Pennsylvania State University

University Park, PA 16802Email: mfm@mara.me.psu.edu

ABSTRACTThe hybrid finite volume/PDFMonte Carlo methodhasboth the

advantagesof thefinitevolumemethod’sefficiencyin solvingflow fieldsandthePDF method’s exactnessin dealingwith chemicalreactions.Itis, therefore,increasinglyusedin turbulentreactiveflow calculations.In orderto resolvethesharpgradientsof flow velocitiesand/orscalars,finegridsor unstructuredsolution-adaptivegridshaveto beusedin thefinite volumecode. As a result,thecalculationdomainis coveredby agrid systemwith very largevariationsin cell size. Suchgridspresentachallengefor acombinedPDF/MonteCarlocode.To date,PDFcalcula-tionshavegenerallybeencarriedoutwith largecells,which assurethateachcell hasastatisticallymeaningfulnumberof particles.Smallercellswould leadto smallernumbersof particlesandcorrespondinglylargerstatisticalerrors. In this paper, a particletracingschemewith adaptivetime stepand particle splitting and combinationis developed,whichallows the PDF/MonteCarlo codeto useany grid that is constructedin the finite volume code. This relaxationof restrictionson the gridmakesit possibleto couplePDF/MonteCarlo methodsto all popularcommercialCFDcodesand,consequently, extendexistingCFD codes’capabilityto simulateturbulentreactiveflow in amoreaccurateway. Toillustratethesolutionprocedure,a PDF/MonteCarlocodeis combinedwith FLUENT to solvea turbulentdiffusioncombustionproblemin anaxisymmetricchannel.

INTRODUCTION

It is well known that PDF/MonteCarlo methodcan treatchemicalreactionsandothernonlinearinteractionsmoreaccu-ratelyandwith relativeease(PopeRef.[1]). In hybridnumericalmethods,theflow field (includingvelocities,pressure,turbulentkinetic energy and the dissipationrate of turbulentkinetic en-ergy) is solvedby finite volumetechniques,while scalarssuchasmassfractionof speciesandtemperaturearesolvedby composi-tion PDF/MonteCarlotechniques.Thehybridmethodscombineboththeadvantagesof thefinite differencemethod’sefficiencyinsolvingtheflow field andthePDFmethod’sexactnessin dealing

with chemicalreactions. It is, therefore,increasinglyusedinturbulentreactiveflow calculations.

In thePDF/MonteCarlosimulation,a largenumberof par-ticles is employedto representtherealflow fields. Meanvaluesof scalarsarecalculatedby averagingovermanyLagrangianpar-ticles. The methodrequiresa sufficient numberof particlestobein eachcell at all timesto keepstatisticalnoiselow to ensurereliable meanvalues. The numberof particlesemployedin asimulationhasa first-ordereffect on the overall costof thecal-culation. A uniform numberof particlesper computationalcellthroughouttheflow domainensuresthatcomputationaleffort isusedefficiently; however, this optimal situationis not attainedin most situations. As pointedout by Li and Modest[2], theuseof cells with large variationsin size is the major causeofunbalancednumberof particlesin differentcomputationalcellsin incompressibleflows. In addition,densityvariationscanalsocausecell populationsto becomeunbalancedsince,if equalmassparticlesareused,low densityregionstendto havelessparticlesin a cell. This imbalanceproblemis particularlyseverein ax-isymmetricsystems,in which cells nearthe axis tendto sufferfrom low particlecountsdueto their small volumeandscalarsaredifficult to resolveaccurately, while outsidecellstendto holdtoomanyparticlesandthuswastescomputerresources.In manysuchflows, sharpgradientsof theflow field occurneartheaxiswhereevensmallercells arerequiredto be used,which makesthingsevenworse.

Similar problemshave beenrecognizedin other types ofMonte Carlo simulations. For instance,in direct Monte Carlosimulationsof rarefiedandnonequilibriumgasflows or in thesimulationof plasma,the imbalanceof particlecountshasalsobeenrecognizedas a problem. Kannenberg andBoyd [3] dis-cussedthreepossibleways to overcomeit: direct variation ofparticleweights,variationof time stepsandgrid manipulation.Lapenta[4] usedparticlesplitting andcoalescing(combination)

to control thenumberof particlespercell in his plasmasimula-tion. All thesetechniqueshavebeenusedin PDF/MonteCarlosimulationsalthoughin a slightly differentinterpretation.

In the field of PDF/MonteCarlo simulation,Li andMod-est[2] proposedaparticletracingschemewith adaptivetimestepsplittingandparticlesplittingandcombinationprocedure,whichcanovercomethe problemof unbalancednumbersof particlesin differentcells andcanimprovethecomputationalefficiency.Theyshowedstrongimprovementin numericalperformanceofthenewschemeby applyingit to a simpleplanardiffusionprob-lem. In thatpaper, thevariationof cell sizeis only about13:1andno chemicalreactionsareconsidered.Thus,a moreseveretestis neededto demonstratethenewmethod’s validity in turbulentreactiveflows.

Thisstudyis a furtherstepin thedevelopmentof a compre-hensivehybrid model for turbulentreactiveflows with thermalradiationeffects. To capturethestronggradientsin suchflows,a locally clusteredmeshor evenanadaptivemeshhasto beusedin thefinite volumesolver. Unstructuredgridsnot only providegreaterflexibility indiscretizingcomplexdomainsbutalsoenablestraightforwardimplementationof adaptivemeshingtechniqueswheremeshpointsmay be addedor deletedlocally (MavriplisRef.[5]). Many finite volumecodes,suchasFLUENT [6], al-low theuseof suchunstructuredmeshes.To accommodatethiscapability, particle tracingneedsto be adaptedto unstructuredmeshes. In this paper, a so-calledelement-to-elementparticletracingstrategyfor unstructuredmeshesis presented,which isvery efficient whenusedwith the timestepsplitting procedure.Theparticlesplitting andcombinationprocedureis discussedinmore depth in this paper, the possibleside effects of splittingandcombiningon thestochasticsystemandtherequirementsofpreservingmeanquantitiessuchasmass,scalarsor momentsofscalarswill beinvestigated.

In the following we first describeour element-to-elementparticletracingmethodin a triangularmesh;thenwe reviewtheadaptivetimestepsplittingandmeanestimationin theframeworkof unstructuredmeshes.The particlesplitting andcombinationprocessis investigatednext, followed by a statictest. Finally, aturbulentreactiveflow in an axisymmetricchannelis solvedtodemonstratetheperformanceof themethod.

PARTICLE TRACING IN TRIANGULAR MESHESIn PDF/MonteCarlosolvers,everyparticlechangesits po-

sition in the computationaldomainduring eachtime stepandthehostcell for it needsto be traced. Therearemanypossiblestrategies(Lohner Ref.[7]) for doingthat. Theuseof successiveneighborsearchesturnsout to bethemostsuitablemethodcon-sideringthefact thatparticlesmostlikely stayin thesamecell orsimplymoveto oneof theirneighboringcellsaftera singletimestep,which is especiallytrue in thecalculationherebecauseoftheuseof anadaptivetimestepsplitting technique.

V1

V2

V3

V2

(V1 )

(V3 )(i)

(i)

(i)

(j )

(j )

(j )

cell i

cell jn1

n2

n3

n1

n2

n3

h3

h1

h1

h2

h2 (h3)(i)

(i)

(i)

(j )

(j )

(j )

(j )

(i)

(i)

(i)

(j )

(j )

O1

O2

Figure 1. Illustration of particle tracing in triangular cells

Assumeaparticleresidesin cell with position 1 originallyandmovesto 2 after onetime step. To identify its new hostcell, first threequantitiesarecalculated,

1 2( )1

( )1

; 1 2( )2

( )2

; 1 2( )3

( )3

where ( )1 , ( )

2 and ( )3 arethreenormalvectors; ( )

1 , ( )2 and ( )

3aretheshortestdistancesfrom point 1 to the threesurfacesofcell (seeFig. 1), andthe superscript( ) is usedto denotethequantities’associationwith cell . If all threequantitiesarelessthanunity theparticleis still in cell ; if theirmaximumis greaterthanunity, this impliesthattheparticleleavesof thecell throughthesurfaceon which the maximumis computed.For example,in thecaseshownin Fig. 1, thesecondof theabovequantitiesisthelargestandtheparticleleavesthecell throughsurface2. Theconnectivityinformationof themeshgivesthecell numbernextto that surface,in this case, . The cell pointerfor the particleis updatedto that of the neighboringcell, and the abovethreequantitiesarerecalculatedwith respectto cell . This processis repeateduntil thenewhostcell is found,i. e. , whenall threevaluesarelessthenunity.

We call this strategyanelement-to-elementsearch.In Sub-ramaniamandHaworth’spaper[8], aface-to-facesearchstrategyis used,in which thetime fractionspentin eachcell is tabulated.Suchinformation is neededif a cell-center-basedPDF codeisused(thefield meansarestoredat thecentroidof thecells)andonewantsto distributea particle’s contributionto all the cellsit passesthroughaccordingto its residencetime in eachcell.As will beseenlater, in thevertex-basedschemewith time stepsplitting procedurethat is usedin our currentmethod(the fieldmeansarestoredatverticesof thecells),theelement-to-elementsearchschemeis moreefficient.

ADAPTIVE TIME STEP SPLITTING AND MEAN ESTIMATIONMETHOD

In thePDF/MCmethod,scalarfieldsneedto beupdatedonanEulerianfinitevolumegridusinginformationfromLagrangianparticles,which carrymassaswell asscalarinformation. Therearemanywaysto calculatefield meansfrom theparticles’prop-erties. The simplestway is to tally particlesonly at the endofa whole time step. Field meansof a cell canbe calculatedassimpleaveragesof thepropertiesof all particlesin a cell or themeanscanbecomputedby weighted-avarging of thepropertiesof its surroundingparticles,in which the weight is determinedby somelinear-basisfunctions.Weighted-averagingleadsto lessvarianceandis widely usedin PDF/MonteCarlomethodstoday.Theabovemethodsonly tally particlesat the endof oneentiretime step. As discussedin our previouspaper(Li andModestRef.[2]), if cells with large variationin sizeareused,the timestepis limited by the time scaleof thesmallestcell in the flowfield makingparticletracingvery inefficient. If a largetimestepis taken,aparticlemaypassthroughseveralcellsin asingletimestep.Asaresult,thisnotonly tendstosmoothentheflow fieldbutalsoleadsto badstatisticssincea particlemaypassthroughsev-eralcellswithout leavinganycontributionto them.To overcomethisshortcoming,anadaptivetime-stepis used,in whichaglobaltime stepis determinedby the largestcell time scale,which isthensplit into sub-steps,if necessary, accordingto theparticle’sposition. Here,a time stepis split suchthat a particlechangesits locationneithertoo little nor too muchin a singlemove,bycomparingthemoveto thecell sizein which theparticleresides.

In thecontextof anunstructuredtriangularmesh,theparti-cle’s stochasticdifferentialequations(SDEs)areintegratedfor-wardin eachsub-stepandaparticleexperiencesaseriesof movesin oneglobaltimestepasshownin Fig. 2. Foraparticle in cell

after the ( 1)th moveat sometime level, thesub-stepforthe th moveis computedas,

= min (1)

where arethemaximumspanof cell in the anddirections,and , , aretheturbulentkinetic energy, itsdissipationrate,andtwo velocity componentsat thecentroidofthecell , respectively. After settingthe partial time step,a setof SDEsfor particle needsto beintegrated.Predictor/corrector(P/C) schemeis usedin the integrationto achievebetter timeaccuracy. In the compositionPDF method,a particle’s loca-tion and its propertiesevolve accordingto (SubramaniamandHaworth Ref.[8]),

( ) = [( + ] ( ) + [2 ]12

( ) (2)

( ) =12

( )1

(3)

wherevariableswith anasteriskreferto theLagrangianparticle’sproperties,and is the turbulenttime scalecalculatedas ;

= 1 2 is theturbulentdiffisivity, where , andareturbulentSchmidtorPrandtlnumberandamodelconstant.is an isotropicvectorWienerprocess.In equation(3) Dopazo’sscalarmixing modelhasbeenusedwhere is anothermodelconstant.Finally, is asourcetermdueto chemicalreactions.

V

A

B

C

D

X

Y

ver tices of elements (nodal points)

dependence region of V

par ticle’s path

V1

V2

V3

O

Cell ithree influence nodes ofany par ticle in cell i

Figure 2. The unstructured mesh used in the hybrid FV/PDF method

Thepredictorstepdeterminestheparticle’sapproximatelo-cationaftertheintegration,with theexplicit form,

+1 = + [( + ] + [2 ]12 (4)

superscripts and + 1 denotevaluesat time and +1 =+ , respectively, and is a standardizedGaussianrandom

vector (componentswith zero meanand unity variance). Allmeanquantitiesappearingin thecoefficientsareevaluatedat theparticle’s initial position as indicatedby the subscript. Inour calculations,all meanvaluesarestoredat theverticesof thetriangularmesh. The field meansat arbitrary particle position

arelinearly interpolatedfrom thethreevertexquantitiesof itshostcell: if denotesanymeanquantitystoredat vertices,thecorrespondingvalueatanarbitrarypoint is,

= 1 1 + 2 2 + 3 3 (5)

where 1, 2 and 3 arethemeanquantitiesatthethreevertices,andtheweightfactors arecalculatedfromalinearbasisfunction.If we confineour discussionto 2D triangularmeshes,theseare,

1 = 2 3

1 2 3

2 = 1 3

1 2 3

3 = 1 2

1 2 3

(6)

where 1 2 3, etc.,aretheareaof triangle 1 2 3 andits sub-trianglesasshownin Fig. 2.

In thecorrectorstep,all coefficientsinvolving thefieldmeansareevaluatedastheaverageof theirmeanvaluesattheinitial andat thepredictedlocation.Thecorrectorstephastheform,

+1 = +12

[ + ] + [ + ] +1 +

1

2[ ]

12 + [ ]

12

+1 (7)

+1 = ++ +1

2 + +1

(8)Notethatthesamevalueof is usedfor bothpredictorandcor-rectorsteps,but that differentand independentvaluesof areusedfor eachsubtimestep. It hasbeenverified that this P/Cschemeis equivalentto the Runge-Kuttaform of Milischtein’sgeneralsecond-orderweakschemefor stochasticsystems(Wel-ton Ref.[9]).

Themeanquantitiesatnodesarecalculatedfromall particlesin a region(dependenceregion)boundedby surroundingnodes.Theyareupdatedat theendof eachglobaltimestepusing,both,the particlespropertiesat the final position as well as thoseatintermediatepositions. The contributionof eachsub-moveisweightedby its time fraction,sothemeanfield of scalarscanbecalculatedas

=1

=1

; ==1

(9)where isasetof particlesthatfall into thedependenceregionofpoint ; is thenumberof sub-stepsfor particle takingit fromtime level to time level + 1; arethescalarvaluesof theth particleat the th substep; is thenormalizedmassthatthe

particlecarries(if everyparticlecarriesthesameamountof mass,then = 1 0); is the weight of the particle’s contributionto point , and is obtainedfrom the linear basisfunction inequation(6).

Strictly speaking,the mean estimationmethod of equa-tion(9) is only valid for statisticallystationaryproblemssincetheparticle’spropertiesat someintermediatetime pointsotherthanat( + ) areusedaswell. But for nonstationaryproblem,if theglobal time stepis much lessthan the characteristictime scaleof theflow field, this averagemethodis still valid. If theglobaltime stepis not that small, thefield meanscanbecalculatedbyonly tallying particlesat theendof oneglobaltimestep,or

=1

; = (10)

PARTICLE SPLITTING AND COMBINING PROCEDUREIn order to achievea uniform statisticaluncertaintylevel

throughoutthe computationaldomain, it is important to havewell-balancednumbersof particlesresidingat all timesin eachcomputationalcell (which may vary vastly in size). A particlesplittingandcombinationprocedureis usedfor thispurpose.Af-tereveryglobal timestep,thenumberof particlesin eachcell ischecked.If the numberin a cell is lessthana prescribedmini-mum,a particlesplitting processis initiated,andif the numberof particlesexceedsthe prescribedmaximum,a particlecombi-nationprocessis carriedout. During this process,a new setofparticlesreplacedthe old. The replacementsethasa differentnumber, but the new particlesshouldcorrespondto the samedistributionof theold field in a statisticalsense.

Theproblemcanbegeneralizedmathematicallyasfollows.Given particlesin the dependenceregion of node withmasses 0, scalars andpositions , a secondsetofparticlescanbegeneratedwith masses 0, scalars andpositions suchthatit preservesthefield variables,i.e.,mass,scalars,scalarvariancesandotherhigher-ordermoments.

= =1

=1

= =1

=1

(11)

= =1

=1

= =1

=1

(12)

2 = =12

=1

= =12

=1

(13)

...

After splitting, thenewly-createdparticlesalwayscarry thesamepropertiesandareputat thesamelocationastheirparents.Theyequallysharetheoriginal massof their parents.Thecon-servationrelationsarealwayspreserved.However, thesplittingprocesscanchangetheevolutionof thestochasticsystemafter-wards,sincesplittingproducesseveralidenticalparticlesin acellandahighdegreeof datadependencymayleadto biasedresults.This would be the case,for example,if the particle-interactionmixing model (Pope Ref.[1]) is usedin the particleevolutionprocess.If everyparticleevolvesindependentlyfrom others,thisdatadependencywill not causeany statisticaldegradation. Inour currentnumericalimplementation,Dopazo’s mixing modelis used,in whichparticlesonly interactwith themeanfieldsand,therefore,splitting doesnot introduceanysideeffects.

The biggestconcernwith the combinationprocessis thatmeanscalarsandtheirmomentsneedto bepreserved.However,equations(11)through(13)arenontrivial tosolve.Therefore,we

look asimplerproblem:if two particles( 1, 1, 1 and 2, 2,

2) in cell arecombinedinto one,whatpropertyshouldthenewparticletakeandwhereshouldit be put in orderto preservethefield means?To preservetheflow field, the following relationsneedto besatisfied,

mass: 1 1 + 2 2 = 0 0 (14)

scalar: 1 1 1 + 2 2 2 = 0 0 0 (15)

others:...

wheresubscript0 indicatesthe new particleand denotesoneof threeinfluenceverticesof particles.Theaboverequirementsaresufficient but not necessaryconditionsto preservethe fieldvalues. Even if they are not satisfied,equations(11) through(13)canstill besatisfiedaftercalculatingtheaverageovermanycombinedpairs.

Wewantto seekthepossiblesolutionfor thisstrongform ofanequationset. If two particlesat ( 1 1) and( 2 2) arecom-bined,we want to calculatethenew particle’s location( 0 0),mass 0, and its property( 0) so that field meansbeforeandafterthecombinationarethesame.

To preservethemass,thenewparticlemustsatisfy

1 1 + 2 2 = 0 0 = 1 2 3 (16)

To preservethescalar, it is requiredthat

1 1 1 + 2 2 2 = 0 0 0 = 1 2 3 (17)

In this setof 6 equationstherearefour unknowns, 0, 0, 0,and 0. Exceptfor somespecialcasesthewholesetof equations,strictly speaking,cannot be satisfied.The first threeequationsaremost importantandwe requirethemto be satisfiedexactly,whichresultsin

0 = 1 + 2 (18)

0 = 1 1 + 2 2

1 + 2

(19)

0 = 1 1 + 2 2

1 + 2

(20)

Thescalarcannotbepreservedexactlyfor everynode.Howeveradding equations(17) leadsto (Using 1 + 2 + 3 = 1,

= 0 1 2),

0 = 1 1 + 2 2

1 + 2(21)

If the new particle’s locationandpropertiesarecalculatedby equations(18)through(21),massis exactlypreservedandthescalaris preservedin an averagesense.So far, we havenot putanyrestrictionon choosingtheparticlesfor combination.If twocombinedparticleshaveidentical properties( 1 = 2), equa-tions(17)arereducedto (16)andtheabovecombinationprocesspreservesthescalarat thenodesexactly. Thisobservationgivesusat leastaguidelinefor choosingparticleswhencombining:inthecombinationprocess,it isbesttochooseparticleswith similarproperties.In complexproblems,we might havemanyscalars,in whichcaseonemustdeterminewhichscalaris mostimportantwhenchoosingthecombinedparticles.

So far, we seethe combinationprocesscan preservethemassandscalarfield by judiciously choosingthe particles. Intherealsimulation,thehighermomentsof theflow field arealsodesiredto be preserved.Obviously the presentmethodmakesthevarianceof scalarsdecreasingjust like thesituationwhenonereplacestwonumbers,suchas,5and-5with 0,althoughthemeanremainsthesame,thestatisticsaretotally different.To preservethehighermomentsof thescalar, saythefirst moment,thestrongform requires,

121 1 + 2

22 2 = 0

20 0 = 1 2 3 (22)

This cannotbe solvedby just usingtwo original particles’information,becausethe field meansarerequiredfor the com-putationof 2 , which needsall the particle’s propertiesnearthat node. Recall that equations(22) areonly sufficient but notnecessaryconditionsto satisfyconservationrequirements.Sinceit is only necessaryto satisfyequations(13) in anaveragesense,twodifferentapproacheshavebeenattemptedtopreservethefirstmomentsof scalars.

In the first approach,the propertyof the new particle iscalculatedas,

0 = 1 1 + 2 2

1 + 2

+12

( 1 2) (23)

where is a standardizedrandomnumber. Theadditionaltermdoesnot affect conservationof massnor themeanscalar, but itcompensatesfor theeffect of decreasingthescalar’s variance.

A second,evensimplermethodto preservethemass,scalarand the first momentof the scalarof the meanfield can alsobeused:whentwo particlesarecombined,ratherthancalculatethenewparticle’spropertiesfrom thepropertiesof thecombinedparticles,onecanjustassignnewpropertiesto it suchas

0 =0+ 2

0

(24)

The particle’s information before combinationis lost but thenewly assignedpropertypreservesthe scalarand its first mo-mentof thefield means.

In our actualsimulations,all particlesin eachcell aresortedin descendingorderof thechosenscalarquantity(temperatureormassfractionof acertainspecies).If thenumberof particlesin acell is largerthantheprescribedmaximumby 2 particles,theseparticlesarecombinedinto pairs.

To checkthe possibledegradationof statisticsof the fieldvariableswhencombinationis invoked,a simplestatictesthasbeenconducted,in which particlesdo not move and the soleeffect due to the particle combinationis separated.Initially , alargenumberof particlesisdistributedoverarectangulardomain,which is coveredby varioussizesof triangularelements,andthescalarthateachparticlecarriesis assignedas = 100+ 10 ,where is a standardizedrandomnumber. Thusthemeanvalueof the scalaris 100, and its standarddeviationis 10. A targetnumberof 10 particlesis set for eachcell. If the numberofparticlesin a cell exceedsthat number,thecombinationprocessis employed.Theinitial particlenumberdensityis madesolargethat it takesseveralcombinationprocessesuntil the numberofparticlesin eachcell dropsbelowtheprescribedtarget.

Two differentsimulationshavebeenperformedwith a totalnumberof 50,000particles.In thefirst one,all particlesinitiallycarryequalmass.Dependingon thecell size,particlesarethencombinedagainandagain. To assesstheeffect of combinationprocesson theparticleswith unequalmass,a secondsimulationis donein which everyparticlepicksup its massrandomlyfromtherange(0,1). In bothsimulations,threecombinationstrategieshavebeenused,calculatingthenewparticle’spropertyfrom,

( ) 0 = 1 1 + 2 2

1 + 2

(25)

( ) 0 = 1 1 + 2 2

1 + 2

+12

( 1 2) (26)

( ) 0 =0+ 2

0

(27)

Themeanof theflow field andits variancearethencalculated,

= =1

=1

2 = =12

=

where is thenumberof nodesin thecomputationaldomainandand 2 areagainmass,meanscalarandits variance

at thenodepoint , which arecalculatedby equation(10).Thenodevaluesareaveragedovertheflow field andresults

areshownin Table1, indicatingthatall combinationapproachespreservethe scalarmeanvery well, but in the first approach

thescalar’s deviationis decreasingunphysicallyasparticlesarecombined,while it is preservedin the secondandthe third ap-proach. In our combustiontestproblem,the third approachhasbeenadopted.

COMBUSTION TEST PROBLEMA numericalsimulationof a cylindrical combustoris made

usingourhybridFV/PDFMonteCarlomethod.Thegeometryofthecombustoris shownin Fig. 3. A smallnozzlein thecenterofthecombustorintroducesmethaneat 80m/s.Ambientair entersthecombustorcoaxiallyat0.5m/s.Theoverallequivalenceratiois approximately0.76 (about28% excessair). The high-speedmethanejet initially expandswith little interferencefrom theouterwall, andentrainsandmixeswith the low-speedair. TheReynoldsnumberbasedon the methanejet diameteris approx-imately 28,000. The flow field quantities(velocities,pressure,turbulentkinetic energy andthedissipationrateof turbulentki-neticenergy) areprovidedby acommercialsolver, FLUENT [6],andspeciesandtemperaturefieldsarefoundwith our composi-

Table 1. The effect of combination process on scalar values and itsdeviations (a) initially using equal mass particles (b) initially using particleswith random mass

(a)

Totalnumber scalarmean scalardeviation

of particles a b c a b c

50000 100.1 100.1 100.1 9.98 9.98 9.98

25100 100.1 100.1 100.2 7.80 10.7 9.98

12715 100.1 100.1 100.2 5.75 10.9 10.0

6665 100.1 100.1 100.1 4.14 10.7 9.93

3850 100.1 100.1 100.2 3.10 10.6 9.98

2931 100.1 100.2 100.3 2.64 10.2 9.64

(b)

Totalnumber scalarmean scalardeviation

of particles a b c a b c

50000 100.0 100.0 100.0 9.96 9.96 9.96

25092 100.0 100.1 100.1 7.00 9.99 9.92

12705 100.0 100.1 100.2 5.00 10.0 9.96

6649 100.0 100.1 100.2 3.60 9.97 10.1

3846 100.0 100.2 100.3 2.69 9.86 9.95

2929 100.0 100.2 100.3 2.31 9.80 9.86

tion PDF solver. The two solversarecloselycoupledsincethePDF solver needsflow field information from the flow solver,andthe flow solver, on the otherhand,needsthe densityfieldcomputedby thePDFsolver. Consequently, thetwo codeshaveto run hand-inhandinteractivelyfor every iteration. Onekeyissuein thehybridschemeis thepassingof informationbetweenthesetwo codes.ThePDF/MonteCarlotechniqueis astatisticalmethodand,hence,alwayshasfluctuationsassociatedwith itssolutions. If the level of thesefluctuationsis too high, it maycausedivergenceor othercomputationaldifficultieswheninfor-mationis communicatedto thefinite volumesolver. Theapplica-tion of our adaptivetime stepsplitting andparticlesplitting andcombinationkeepsstatisticalfluctuationslow andno divergenceproblemwasobservedin thefinite volumecodeaftertheupdateddensityfield was fed back from the PDF solver, evenfor lowparticlenumberdensities,say10particlespercell in average.Toacceleratethecalculation,thesolutionproceedsasfollows: firsttheturbulentreactiveflow problemis solvedby FLUENT usingtheeddy-dissipationcombustionmodel(whichrelatestherateofreactionsto the rateof dissipationof the reactant-andproduct-containingeddies);thenthesteady-statesolutionof thevelocityfield obtainedby FLUENT is fedto thePDF/MonteCarlosolver,whichfindsthestatisticallysteadystatesolutionof scalarsfor thatgivenvelocity field; a newdensityfield is fed backto FLUENTandtheprocessis iterateduntil thefinal steadystatesolutionisreached.

Air , 0.5 m/s, 300K

Methane, 80m/s, 300K

1.8m

0.225m

0.005m

q=0

Figure 3. Geometry of the cylindrical combustor

Two setsof unstructuredmeshes,onewith 1247nodesand2317cellsandtheotherwith 4753nodesand9103cellswereuseddemonstratinggrid-independenceof the finite volumesolution.The coarsermeshis usedfor the solution of this problem inthehybrid finite volume/PDFMC calculationwhich is showninFig. 4. Sincethediameterof theinnerfuel nozzleis very small,a lot of cells haveto clusternear the fuel inlet to adequatelycapturetheflow field. As aresult,theminimumcell areais only7 3 10 7m2 while themaximumcell areais 1 4 10 3m2. Incylindricalcoordinates,acell in the - planeactuallyrepresentsanannulusin threedimensionalspace.Forthisproblem,thetotalvolumeof thecomputationaldomainisabout4 55 10 2m3, and

theminimumandmaximumcell volumesare4 0 10 10m3 and2 0 10 4m3, respectively, or:

min

total= 8 8 10 9; max

total= 4 4 10 3; max

min= 5 105 (28)

In the traditionalPDF/MC method,particlesof equalmassareusedthroughout. If 1,000,000,000particleswere usedin thesimulation,thelargestcell alonewould hold 4,400,000particleswhile the smallestcell would hold only about9 particles. Ob-viously, thetraditionalmethodis extremelyinefficient in tracingparticlesin aproblemwith suchunbalancedcontrolvolumes.

Figure 4. The unstructured mesh used in the hybrid FV/PDF Monte Carlomethod

While solving this kind of problemby traditionalmethodsmay still be possible,althoughvery inefficient, with a supercomputer, it becomescompletelyimpossiblewhentherequiredtime stepis considered.A particleenteringwith thefuel streamtakesabout0.0225sto travel throughthe entire computationaldomain,while aparticlefrom theair streamwill takeabout3.6s.In conventionalmethods,aconstanttimestepfor all theparticlesisused,whichisconstrainedby thesmallestcell regionwherethevelocity is thelargest.If sucha timestepis used,particlesin thelargeoutsidecellswouldmoveonly atiny distancein eachsingletimestepandthesimulationwould essentiallytakeforever.

Our new particle tracing schemewith adaptivetime stepsplitting and with the particle splitting and combinationtech-niquecanhandlethis extremecasewithout any problems. Thenumberof particlesin thecomputationalfield is about46,500inthesimulation,andtheparticlesplitting andcombinationproce-dureensuresthat thenumberof particlesin eachcell preferablyremainsin therangeof 10-50.Thescalarquantitiesin thisprob-lem includefive massfractionsof different species(CH4, O2,CO2, H2O andN2) andthe temperatureT. A particlewith hightemperaturewill most likely also havehigh massfractionsofCO2 andH2O. To preserveasmanyscalarsaspossible,in thecombinationprocess,we chooseto combineparticleswith sim-ilar temperature.To reducethestatisticalerror, thetemperatureandspeciesfieldsin thePDFsolverareobtainedfrom averagingtheirsolutionsovertheprevious5 timesteps.In theentiresimu-lation,theglobaltimestepof 0.004sis usedand1000globaltimestepsaretakento reachthesteadystatesolution,takingabout50

minuteson anSGI-0200(180MHz,singleR10000processor)tocompletethewholesimulation.

The conventionalway to define the residualerror in thefinite volumemethodis meaninglessin thehybrid FV/PDFMCsimulationbecausethe statisticalerror is generallylarger thanthe truncationerror. The numericalerror for any variable isdefinedhereas

err =1

=1

[ ( + ) ( ) ]2

( ) 2(29)

This error neverconvergesto zero,but ratherto a valuerepre-sentativeof thestatisticalfluctuationof thesolutionwhensteadystateis reached. This level mainly dependson the numberofparticlesin thesimulationandon thenumberof time stepsoverwhich thesolutionis averaged.If temperatureis usedto monitorthenumericalerror, avalueontheorderof 10 4 hasbeenreachedin thepresentcalculations.

Duringeveryglobaltimestepsomeparticlesmightpasssev-eralcells througha seriesof sub-moveswhile othersmight justtakeonemovedependingon their locationin thecomputationaldomain. During eachtime stepa numberof particlesleavesthecomputationaldomainand,at the sametime, a similar numberof new onesareintroducedat the inlet to maintainglobalmassconservation.Thenewparticlesarereleasedsuchthat themassflow rate is constantfor the fuel and air streamsand their ra-tio is equalto thegiven equivalenceratio of the fuel. AlthoughPDF/MonteCarlomethodscantreatchemicalreactionsexactlyifcorrectchemicalkinetic relationsareused,here,for thepurposeof comparisonwith FLUENT, aninfinitely fastchemicalreactionrateis usedfor everyparticle,implying thatthemacro-chemical-reaction-rateis controlledby themixing process,whichis similarto the eddy-dissipationmodelusedby FLUENT. In the simula-tion,althoughtheparticlenumberdensitymaychangedueto thesplitting andcombinationprocedure,theparticle’s massdensitylevelisalwaysproportionaltotheflow fielddensity. Thecontoursfor themassfractionof CH4, O2, CO2 andtemperatureareshownin Fig. 5 throughFig. 8 (Themassfractionof H2O hasthesameprofile asthat of CO2 whenscaledproperly, andis not shown).Theresultsfrom FLUENT with theeddy-dissipationcombustionmodelarealsoshownfor comparison.ThePDF/MCmethodandFLUENTgiveverysimilarscalarfields. Theslightdiscrepanciesaredueto severalsources.Firstly, FLUENT’s resultsdependonthespecificcombustionmodelchosenin thesimulationandarealwaysapproximate.Althoughtheinfinite reactionrateassump-tionhasbeenusedin thePDF/MonteCarlocalculationto createasimilarsituationasthatin FLUENT, two combustionmodelsarenot exactlythesame. Secondly, the simplemixing modelusedin our PDF/MonteCarlo methodalso leadsto somenumericalerrors.Finally, PDF/MCresultsalwayshavefluctuationsandthenumberof particlesusedin our simulationis quite low for this

complexproblemandamuchsmoothersolutionscanbeexpectedif moreparticleswereemployed. While further improvementscanbemadein theseareas,themainfocusin thisstudyis on thedevelopmentof asolutionprocedure.

1.000.930.860.790.710.640.570.500.430.360.290.210.140.070.00

PDF/MC results

FLUENT results

Figure 5. The contours of mass fraction of CH4

0.230.210.200.180.160.150.130.120.100.080.070.050.030.020.00

PDF/MC results

FLUENT results

Figure 6. The contours of mass fraction of O2

0.140.130.120.110.100.090.080.070.060.050.040.030.020.010.00

PDF/MC results

FLUENT results

Figure 7. The contours of mass fraction of CO2

CONCLUSIONTo capturesharpgradientsin a flow field, cell systemswith

largevariationsin sizehaveto beused,which causeunbalanced

30002807261424212229203618431650145712641071879686493300

PDF/MC results

FLUENT results

Figure 8. The contours of temperature

numberof particle in different computationalcells. This bal-anceis especiallyseverewhenusingcylindrical coordinatesinaxisymmetricproblems.

A newPDF/MonteCarlomethodwasdevelopedfor unstruc-turedmeshes,whichallowstheuseof arbitrarycellsizevariationssuchastheonesusedin finite volumecodes.In thisnewmethod,anadaptivetimestepsplittingandaparticlesplittingandcombi-nationprocessareintroduced,which leadto near-equalparticlecountsin eachcell. By judiciouslychoosingcombinedparticles,the combinationprocedurepreservesscalarsandtheir first mo-mentsof themeanfield. As a test,a cylindrical combustorwassimulatedwith this hybrid FV/ PDF MonteCarlo method. TheconventionalPDF/MonteCarlomethodcannothandleaproblemof suchcomplexity, while it is dealtwith, with ease,usingournewmethod.

ACKNOWLEDGMENTSThis work is fundedby theNSF/SandiaNationalLaborato-

ries life cycleengineeringprogramthroughgrantnumberCTS-9732223.

REFERENCES[1] Pope,S. B., 1985, “PDF Methodsfor TurbulentReactiveFlows”, Progressin Energy and CombustionScience, 11, pp.119–192.[2] Li, G. andModest,M. F., 2000,“An Efficient ParticleTrac-ing Schemein PDF/MonteCarlo Methods”,Acceptedby 34thNationalHeatTransferConference.[3] Kannenberg, K. C. and Boyd, I. D., 2000, “StrategiesforEfficientParticleResolutionin theDirectSimulationMonteCarloMethod”,J. Comput.Phys., 157, pp.727–745.[4] Lapenta,G., 1994,“Dynamic andSelectiveControl of theNumberof Particlesin Kinetic PlasmaSimulations”,J. Comput.Phys., 115, pp.213–227.[5] Mavriplis, D. J., 1997, “UnstructuredGrid Techniques”,Annu.Rev. Fluid Mech., 29, pp.473–514.[6] FLUENT, 1998,“ComputationalFluidDynamicsSoftware”,Version5.

[7] Lohner, R.,1990,“A VectorizedParticleTracerfor Unstruc-turedGrids”, J. Comput.Phys., 91, pp.22–31.[8] Subramaniam,S.andHaworth,D. C.,1999,“A PDFMethodfor TurbulentMixing andCombustiononThree-DimensionalUn-structuredDeformingMeshes”,(submittedto Journalof EngineResearch).[9] Welton,W. C. andPope,S. B., 1997,“Model Calculationsof CompressibleTurbulentFlowsUsingSmoothedParticleHy-drodynamics”,J. Comput.Phys., 134, pp.150–168.